Electromagnetism - Lecture 9
Dielectric Materials
• Electric Dipole Moments of Atoms
• Polarization Vector P
• Local Electric Field
• Electric Displacement Vector D
• Linearity, Isotropy and Homogeneity
• Polar Molecules
1
Polarizability of Atoms
In an insulator the electrons are bound to the atomic lattice and
cannot move freely ... but in the presence of an external electric
field E each atom becomes slightly polarized
The atomic dipole moment is:
p = Zea = α0 EL
where the separation between the centroid of the electron cloud
and the atomic nucleus is small compared to the Bohr radius:
|a| ≈ 10−15 m = 10−5 a0
α is known as the atomic or molecular polarizability
EL is the local electric field inside the material which is not
identical to the external field E
2
Polarization Vector & Susceptibility
The atomic dipole moment can be expressed as in integral over the
atomic charge density:
Z
rρdτ
p=
atom
The dipole moment density is the polarization vector P:
P=
dp
=< rρ >atom
dτ
The polarization vector is proportional to the external field E:
P = χE 0 E
where χE is the electric susceptibility of the material
3
Notes:
Diagrams:
4
Polarization Flux
The polarization flux through a surface has units of charge:
d(p.n̂)
= −QP
dl
The polarization charge QP is equivalent to induced surface charges
σP = ∓|P| on the surfaces perpendicular to the external field
P.dS =
Integrating the polarization flux over a closed surface
Z
I
I
d(p.n̂)
=−
P.dS =
ρP dτ
dl
A
A
V
where ρP is the polarization charge density
Using the divergence theorem:
∇.P = −ρP
The distributions σP and ρP represent the effect of the atomic
polarization with equivalent macroscopic charge distributions
5
Local Electric Field
Inside the material the polarization vector P modifies the local
electric field EL compared to the external electric field E
P
χE E
= 1+
EL = E +
30
3
The factor 1/3 will be derived in a later lecture
The polarization vector can be expressed microscopically in terms
of EL and macroscopically in terms of E:
P = NA α0 EL
P = χE 0 E
where NA is Avogadro’s number and α is the atomic polarizability
χE χE = N A α 1 +
3
χE =
For small polarizability NA α 1:
6
NA α
(1 − NA α/3)
χE ≈ NA α and EL ≈ E
Notes:
Diagrams:
7
Notes:
Diagrams:
8
Electric Displacement Vector
Gauss’s Law is modified to include polarization effects:
I
Z
1
(ρC + ρP )
E.dS =
(ρC + ρP )dτ
∇.E =
0 V
0
A
where ρC are free charges (if any).
Using ∇.P = −ρP this can be rewritten as:
∇.P
ρC
−
∇.E =
0
0
∇.(0 E + P) = ρC
The electric displacement vector D is:
D = 0 E + P
∇.D = ρC
Note that D and P have units Cm−2 , whereas E is in V m−1
9
Dielectric Constant
The linear relationships between D, E and P:
D = 0 E + P
P = χ E 0 E
can be expressed in terms of a dielectric constant r
D = r 0 E
r = 1 + χE
For most dielectrics χE 1 and r is slightly above 1
This reflects the small atomic polarizability α of most materials
General advice - wherever 0 appears in electromagnetism,
it should be replaced by r 0 for insulators
10
LIH Dielectrics
Implicit in the idea of a dielectric constant for a material are
certain assumptions:
• Linearity - the polarization is proportional to the magnitude of
E. This breaks down at high electric fields.
• Isotropy - the polarization is independent of the direction of E.
Some crystals are anisotropic, particularily under mechanical
stress. Known as the piezoelectric effect.
• Homogeneity - the polarization is the same everywhere in the
material.
Materials satisfying these conditions are known as LIH dielectrics
Most insulators are LIH dielectrics
11
Notes:
Diagrams:
12
Polar Dielectrics
Some molecules have an intrinsic dipole moment p =ea, where
|a| ≈ 10−10 is the molecular size. These polar dielectrics have large
dielectric constants.
An example is water with r = 81 at room temperature
In the absence of an external field E the molecular dipoles are
randomly oriented and P = 0
In the presence of an external E the molecular dipoles align with
the field to generate a large P
13
Susceptibility of Polar Dielectrics
The alignment of the dipoles with the external field is disrupted by
thermal motion:
N (θ)dθ ∝ e−U/kT sin θdθ
U = −p.E = −pE cos θ
Expanding the exponent under the assumption that U kT and
integrating over the orientation angle θ:
NA |p|2
E
P=
3kT
The susceptibility χE decreases with increasing temperature
2
|p|
χE = N A α +
30 kT
where the first term is the usual LIH dielectric result, and the
second term applies to polar dielectrics.
14
Energy Storage in Dielectrics
The capacitance of a capacitor increases if the gap is filled with a
dielectric material:
C = r C0
Hence the energy stored in a capacitor increases:
1
U = CV 2 = r U0
2
The energy density of the electric field also increases:
1
1
dUE
2
= r 0 E = D.E
dτ
2
2
These are very large effects for polar dielectrics...
a lot of electric energy can be stored in a water bath!
15
Notes:
Diagrams:
16